Chapter Review

Letter and Symbols Series

Letter, Number and Symbol Series

Arithmetic Number Series

A sequence where each term changes by a fixed common difference d. The most common and easiest series type to identify.

Key Points

  • •
    Find d by subtracting any term from the next: d = a₂ − a₁
  • •
    Positive d → series rises; negative d → series falls; d = 0 → constant
  • •
    The nth term formula uses (n − 1) multiplications of d, not n
  • •
    Always verify d across at least two consecutive pairs before concluding the pattern is arithmetic
  • •
    First-level differences being constant confirms an arithmetic pattern
Formula

Sum of an Arithmetic Series

The total of the first n terms can be found without adding each term individually by averaging the first and last term.

Key Points

  • •
    Pair the first and last terms — they sum to the same value as the second and second-last
  • •
    There are n/2 such pairs, giving the compact formula
  • •
    Works for both even and odd n (the middle term is automatically accounted for)
  • •
    First find aₙ using the nth-term formula, then apply the sum formula
Formula

Geometric Progression

A sequence where each term is obtained by multiplying the previous term by a fixed common ratio r. Growth is multiplicative rather than additive.

Key Points

  • •
    Find r by dividing any term by its predecessor: r = a₂ / a₁
  • •
    |r| > 1 → terms grow; 0 < |r| < 1 → terms shrink toward zero
  • •
    Negative r causes sign alternation (e.g. 3, −6, 12, −24, …)
  • •
    r can be a fraction (e.g. 81, 27, 9, 3, … has r = 1/3)
  • •
    The exponent in the nth-term formula is n − 1, not n
  • •
    If numbers grow/shrink by increasingly large jumps, suspect geometric rather than arithmetic
Formula

Sum of a Geometric Series

Adds all terms up to a given position using the first term and common ratio. Undefined when r = 1 (use Sₙ = n · a₁ instead).

Key Points

  • •
    When 0 < r < 1, rewrite as Sₙ = a₁ · (1 − rⁿ)/(1 − r) to keep the numerator positive
  • •
    When r = 1, every term equals a₁ so Sₙ = n · a₁
  • •
    Derived by multiplying Sₙ by r and subtracting — most terms cancel
  • •
    Only valid for finite sums (n terms)
Formula

Two-Step and Quadratic Patterns

When first-level differences are not constant, compute second-level differences. A constant second-level difference reveals a quadratic (two-step) pattern.

Key Points

  • •
    First-level differences: subtract each term from the next
  • •
    Second-level differences: subtract each first-level difference from the next
  • •
    If second-level differences are constant → quadratic pattern (e.g. 2, 5, 10, 17, 26 with second-level d = 2)
  • •
    Alternating series may switch between two different operations on odd/even positions (e.g. +3, ×2, +3, ×2)
  • •
    Always check second-level differences before concluding a pattern is non-arithmetic

Special Number Patterns

Series built from squares, cubes, Fibonacci sums, or products of consecutive integers. Quick recognition of these templates saves time.

Key Points

  • •
    Square numbers: 1, 4, 9, 16, 25, … — each term is n²
  • •
    Cube numbers: 1, 8, 27, 64, 125, … — each term is n³
  • •
    Fibonacci-style: each term = sum of previous two (e.g. 2, 3, 5, 8, 13, …)
  • •
    Product pattern: n(n+1) gives 2, 6, 12, 20, 30, …
  • •
    Constant differences → arithmetic; constant ratios → geometric; sum of prev two → Fibonacci
  • •
    Very rapid growth suggests factorial (n!) or exponential patterns

Letter Series and Alphabet Mapping

Convert letters to their alphabet positions (A = 1 through Z = 26) to transform a letter series into a number series, then apply number-pattern rules.

Key Points

  • •
    A = 1, B = 2, …, Z = 26 — memorise at least the vowels: E = 5, I = 9, O = 15, U = 21
  • •
    Reverse position: Z = 1, Y = 2, …, A = 26 — some patterns use mirror positions
  • •
    Vowel/consonant classification can be part of the rule (vowels: A, E, I, O, U)
  • •
    Write positions below the letters to make the pattern visible at a glance

Letter Shifts and Skip Patterns

Letters move forward or backward through the alphabet by a fixed or increasing amount. After Z the alphabet wraps back to A.

Key Points

  • •
    Constant shift: e.g. B, E, H, K, N, … — each letter is +3 positions from the previous
  • •
    Increasing shift: e.g. A, C, F, J, O, … — jumps grow by 1 each time (+2, +3, +4, +5, …)
  • •
    Wrap-around: after Z comes A — e.g. X, Z, B, D, … is +2 with wrap-around
  • •
    Skip of k means k letters are skipped between terms: skip-1 → A, C, E, G, … (every other letter)
  • •
    Count the gap between gaps to detect an increasing shift

Symbol Series

Uses shapes, arrows, or symbols instead of letters/numbers. The rule may involve rotation, element addition/removal, size change, or shading.

Key Points

  • •
    Rotation: e.g. arrow rotates 90° clockwise each step (up → right → down → left)
  • •
    Element addition: a figure gains one line, dot, or shading each step
  • •
    Interleaved symbols: two or more symbols alternate in a cycle (★, ●, ★, ●, …)
  • •
    Check one property at a time: orientation first, then count, then size, then shading
  • •
    A single series may change multiple properties simultaneously — don't stop after finding one

Mixed Series

Combines two or more element types (letters, numbers, symbols) in one sequence. Each element type typically follows its own independent rule.

Key Points

  • •
    Split the sequence into separate tracks — one per element type — before finding rules
  • •
    Parallel patterns: A1, C3, E5, G7, … — letters +2 and numbers +2 independently
  • •
    Interleaved sequences: odd positions form one series, even positions form another
  • •
    Sometimes one element encodes information about the other (e.g. letter position = number)
  • •
    With only 4–5 given terms the rule is likely simple (constant difference or ratio)
  • •
    Always verify the predicted rule on every given term before selecting the answer

Formulas

Arithmetic nth Term

Value of the nth term given first term a₁ and common difference d.

Arithmetic Series Sum

Sum of the first n terms by averaging first and last term.

Geometric nth Term

Value of the nth term given first term a₁ and common ratio r.

Geometric Series Sum (r ≠ 1)

Sum of the first n terms of a geometric sequence. For 0 < r < 1, use Sₙ = a₁ · (1 − rⁿ)/(1 − r).